Coming back to the original topic, the answer is: depending on type. While most aircraft with FADEC engines do stop if both channels are dead, I know of at least one aircraft, the Avro RJ series, where they have one FADEC and one backup mode.
On the other hand I do not see the a problem with FADEC only controlling an engine. In the old days of cable, rods and wires, you could also have a complete control failure of an engine.
Dani